As BF is 1/6 of the circumference, measure of angle BAF is 360°/6 = 60°
In triangle ABF, AB = AF and angle BAF = 60°
Hence, ABF is an equilateral triangle.
As length of the cord BF = 2, AB = AE = radius of the circle = 2

Now, as BC and AE are parallel, height of the triangle ACE = AB
Area of the triangle ACE = (base)*(height)/2 = (AE)*(AB)/2 = (2)*(2)/2 = 2

Now, BC is a tangent to the circle at B and parallel to AE, measure of angle BAE is 90°
Hence, measure of angle FAE is (90° - 60°) = 30°
Therefore, area of the sector AFE = (πr²)*(30/360) = πr²/12 = π(2)²/12 = 4π/12 = π/3

Hence, are of the shaded region = (Area of the triangle ACE - Area of the sector AFE) = (2 - π/3)

The correct answer is E.